Table 1: Lifetimes of / = 49 pressure vessels. 1051 4921 7886 10861 13520 1337 5445 8108 11026 13670 1389 5620 8546 11214 14110 1921 5817 8666 11362 14496 1942 5905 8831 11604 15395 2322 5956 9106 11608 16179 3629 6068 9711 11745 17092 4006 6121 9806 11762 17568 4012 6473 10205 11895 17568 4063 7501 10396 12044 The data in Table 1 are the lifetimes (times to failure in hours) of Kevlar epoxy strand pressure vessels at 70% stress level. A commonly used model for times to failure (or survival times) is the Weibull distribution which has the probability density function f (y; ), A ) = Ay4 - 1 Ox exp - (8)* , 720, 920, X20. and the survival function S(y; ), 0) = expwhere y > 0 is the time to failure, A is a parameter that determines the shape of the distribution and A is parameter that determines the scale. Let Y1, . .. , Yn denote the data, with N = 49. If the data are from a random sample of pressure vessels, we assume the Yo's are independent random variables. If they all have the Weibull distribution with the same parameters, their joint probability distribution is N f(y1, . . . , UN; 0, X) = I i=1 (a) For the data in Table 1, fit a Weibull distribution to the data, assuming both A and 0 are unknown: Find MLEs of the two parameters using the Newton-Raphson method with your own R. functions for iterations. (b) Repeat (a) to find MLEs of the two parameters using the two-dimensional optimization method based on the R. function optim (). (c) Repeat (a) to find MLEs of the two parameters using the nonlinear optimization with con- straints method based on the R. function auglag () from the R package alabama. Note: the two parameters must satisfy 0 > 0 and > > 0